Wedgie - Revision history

FloraC: - misinformation. Cleanup. Style

2025-11-24T09:26:08Z

- misinformation. Cleanup. Style

← Older revision		Revision as of 09:26, 24 November 2025
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	{{Beginner\|Plücker coordinates}}		{{Beginner\|Plücker coordinates}}

	A '''wedgie''' is a primarily mathematical object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] ~~vs.~~ [[generator~~]] or of [[equave~~]], which can therefore provide some illuminating information beyond the [[mapping\|mapping matrix]]~~, which specifies a particular equave~~. A wedgie takes the form ⟨⟨…⟨ ''w''<sub>1</sub> ''w''<sub>2</sub> … ''w''<sub>''n''</sub> ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Wedgies can be thought of as a generalization of vals, called ''multivals'', so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.		A '''wedgie''' is a primarily mathematical object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] and [[generator]], which can therefore provide some illuminating information beyond the [[mapping\|mapping matrix]]. A wedgie takes the form ⟨⟨…⟨ ''w''<sub>1</sub> ''w''<sub>2</sub> … ''w''<sub>''n''</sub> ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Wedgies can be thought of as a generalization of vals, called ''multivals'', so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.

	While wedgies ~~historically~~ served as ~~the~~ canonical way to represent temperaments mathematically due to their ~~equave~~-agnostic nature, ~~and are nowadays a fairly common way to do so, they are of limited~~ musical utility - [[~~Mapping~~\|mappings]] would often be of more immediate use to musicians.		While wedgies have served as a canonical way to represent temperaments mathematically due to their generator-agnostic nature, their musical utility is not obvious – [[mapping\|mappings]] would often be of more immediate use to musicians.

	== How to read a wedgie ==		== How to read a wedgie ==
	Following intuitions from [[ploidacot]], one way to characterize temperaments is by how many parts they split important intervals into - for example, the perfect fifth and the octave.		Following intuitions from [[ploidacot]], one way to characterize temperaments is by how many parts they split important intervals into – for example, the perfect fifth and the octave.

	For any n-prime subgroup of a rank-n temperament, there exist a finite number (1 or more) of copies of that subgroup within the temperament. Think of these as "universes" that are connected exclusively by intervals of that subgroup and may be travelled between by using intervals outside the subgroup. For example, a temperament that is [[Ploidacot/Diploid dicot\|diploid dicot]] - dividing the octave into two parts and also dividing the perfect fifth into two parts - will have a 2.3 wedgie entry of 4 (since there are four distinct copies of the 3-limit - the basic 3-limit, offset by a neutral third, offset by a semioctave, and offset by both). Each wedgie entry counts the number of copies of its corresponding subgroup. Because any temperament can be defined by splitting some interval and assigning the parts just interpretations, this is enough to uniquely characterize the temperament.		For any ''n''-prime subgroup of a rank-''n'' temperament, there exist a finite number (1 or more) of copies of that subgroup within the temperament. Think of these as "universes" that are connected exclusively by intervals of that subgroup and may be travelled between by using intervals outside the subgroup. For example, a temperament that is [[Ploidacot/Diploid dicot\|diploid dicot]] – dividing the octave into two parts and also dividing the perfect fifth into two parts – will have a 2.3 wedgie entry of 4 (since there are four distinct copies of the 3-limit – the basic 3-limit, offset by a neutral third, offset by a semioctave, and offset by both). Each wedgie entry counts the number of copies of its corresponding subgroup. Because any temperament can be defined by splitting some interval and assigning the parts just interpretations, this is enough to uniquely characterize the temperament.

	=== Example 1: ~~Meantone {{multival\| 1 4 4 }}~~ ===		=== Example 1: meantone ===
	~~For example, take the~~ wedgie for meantone: {{multival\| 1 4 4 }}. Each entry corresponds to a pair of primes: 2.3, 2.5, and 3.5. The first entry of the wedgie is the ploidacot signatures multiplied together, which in this case is 1, because the octave finds 2 at one step, and the fifth finds 3 at one generator minus one octave (which, since there are no even splits, still counts as 1).		The wedgie for meantone is {{multival\| 1 4 4 }}. Each entry corresponds to a pair of primes: 2.3, 2.5, and 3.5. The first entry of the wedgie is the ploidacot signatures multiplied together, which in this case is 1, because the octave finds 2 at one step, and the fifth finds 3 at one generator minus one octave (which, since there are no even splits, still counts as 1).

	For 2.5, the procedure generalizes, with the entry, 4, being the number of steps 2 and 5 are divided into respectively multiplied together; that is, the number of parts into which the 2.5 subgroup is split. But since we already know 2 is divided into only one octave, this must mean 5 is split into four parts. In fact, 5 is found at four fifths up.		For 2.5, the procedure generalizes, with the entry, 4, being the number of steps 2 and 5 are divided into respectively multiplied together; that is, the number of parts into which the 2.5 subgroup is split. But since we already know 2 is divided into only one octave, this must mean 5 is split into four parts. In fact, 5 is found at four fifths up.
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	For the final entry, which is for the 3.5 subgroup, we have another 4. But this time, we are thinking tritave-equivalently now, so we will be reaching 5/3. This is the number of parts 3 and 5/3 are divided into respectively, multiplied together. 3 is reached by going up one 3/2 and one 2/1, but no splitting is happening, so the factor of 4 must come from 5/3, which is indeed reached by four 3/2's.		For the final entry, which is for the 3.5 subgroup, we have another 4. But this time, we are thinking tritave-equivalently now, so we will be reaching 5/3. This is the number of parts 3 and 5/3 are divided into respectively, multiplied together. 3 is reached by going up one 3/2 and one 2/1, but no splitting is happening, so the factor of 4 must come from 5/3, which is indeed reached by four 3/2's.

	=== Example 2: ~~Father {{multival\| 1 -1 -4 }}~~ ===		=== Example 2: father ===
	For another example, take father, which has the wedgie {{multival\| 1 -1 -4 }}.		For another example, take father, which has the wedgie {{multival\| 1 -1 -4 }}.

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	Finally, for 3.5, we have the entry -4. Again, we are tritave-equivalent and 3/1 is simply found by an octave and a fifth, so we will be finding 5/3 by splitting it into four parts. 5/3 is equated to 16/9 in father, which is found by going up two octaves and down two fifths. This might seem like only a split into two, but keep in mind - we are in tritave-equivalent territory. Octaves are the tritave complement of fifths. So instead of going up two octaves, we can instead simply go down two more fifths to reach 5/3. And there we have it – 5/3 is split into four parts, which each contain a negative generator.		Finally, for 3.5, we have the entry -4. Again, we are tritave-equivalent and 3/1 is simply found by an octave and a fifth, so we will be finding 5/3 by splitting it into four parts. 5/3 is equated to 16/9 in father, which is found by going up two octaves and down two fifths. This might seem like only a split into two, but keep in mind - we are in tritave-equivalent territory. Octaves are the tritave complement of fifths. So instead of going up two octaves, we can instead simply go down two more fifths to reach 5/3. And there we have it – 5/3 is split into four parts, which each contain a negative generator.

	=== Example 3: ~~Blackwood {{multival\| 0 5 8 }}~~ ===		=== Example 3: blackwood ===
	For our final example, we will consider blackwood {{multival\| 0 5 8 }}.		For our final example, we will consider blackwood {{multival\| 0 5 8 }}.

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	For wedgies of higher-rank temperaments, the number of primes per entry is increased, so that for a rank-3 temperament of the 7-limit, all possible combinations of 3 primes (2.3.5, 2.3.7, 2.5.7, and 3.5.7) would be covered.		For wedgies of higher-rank temperaments, the number of primes per entry is increased, so that for a rank-3 temperament of the 7-limit, all possible combinations of 3 primes (2.3.5, 2.3.7, 2.5.7, and 3.5.7) would be covered.

	== ~~The form~~ of a wedgie ==		== Form of a wedgie ==
	The notation being used previously, {{multival\| x y z }}, is formally a shorthand for a matrix form, written		The notation being used previously, {{multival\| ''x'' ''y'' ''z'' }}, is formally a shorthand for a matrix form, written

	$$		$$
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	$$		$$

	For a wedgie on a 4-prime subgroup, the structure of {{multival\| a b c d e f }} is actually		For a wedgie on a 4-prime subgroup, the structure of {{multival\| ''a'' ''b'' ''c'' ''d'' ''e'' ''f'' }} is actually

	$$		$$

VectorGraphics at 05:06, 24 November 2025

2025-11-24T05:06:42Z

← Older revision		Revision as of 05:06, 24 November 2025
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	{{Beginner\|Plücker coordinates}}		{{Beginner\|Plücker coordinates}}

	A '''wedgie''' is an object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] vs. [[generator]] or of [[equave]], which can therefore provide some illuminating information beyond the [[mapping\|mapping matrix]], which specifies a particular equave. A wedgie takes the form ⟨⟨…⟨ ''w''<sub>1</sub> ''w''<sub>2</sub> … ''w''<sub>''n''</sub> ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Wedgies can be thought of as a generalization of vals, called ''multivals'', so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.		A '''wedgie''' is a primarily mathematical object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] vs. [[generator]] or of [[equave]], which can therefore provide some illuminating information beyond the [[mapping\|mapping matrix]], which specifies a particular equave. A wedgie takes the form ⟨⟨…⟨ ''w''<sub>1</sub> ''w''<sub>2</sub> … ''w''<sub>''n''</sub> ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Wedgies can be thought of as a generalization of vals, called ''multivals'', so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.

			While wedgies historically served as the canonical way to represent temperaments mathematically due to their equave-agnostic nature, and are nowadays a fairly common way to do so, they are of limited musical utility - [[Mapping\|mappings]] would often be of more immediate use to musicians.

	== How to read a wedgie ==		== How to read a wedgie ==

VectorGraphics at 04:21, 4 November 2025

2025-11-04T04:21:41Z

← Older revision		Revision as of 04:21, 4 November 2025
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	{{Beginner\|Plücker coordinates}}		{{Beginner\|Plücker coordinates}}

	A '''wedgie''' is an object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] vs. [[generator]] or of [[equave]], which can therefore provide some illuminating information beyond the [[mapping\|mapping matrix]], which specifies a particular equave. A wedgie takes the form ⟨⟨…⟨ ''w''<sub>1</sub> ''w''<sub>2</sub> … ''w''<sub>''n''</sub> ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Wedgies can be thought of as a generalization of vals, called ''multivals'', so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.		A '''wedgie''' is an object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] vs. [[generator]] or of [[equave]], which can therefore provide some illuminating information beyond the [[mapping\|mapping matrix]], which specifies a particular equave. A wedgie takes the form ⟨⟨…⟨ ''w''<sub>1</sub> ''w''<sub>2</sub> … ''w''<sub>''n''</sub> ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Wedgies can be thought of as a generalization of vals, called ''multivals'', so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.

	== How to read a wedgie ==		== How to read a wedgie ==
	Following intuitions from [[ploidacot]], one way to characterize ~~a temperament~~ is how many ~~equal~~ parts ~~it splits~~ the perfect fifth ~~(3/2) into. For an example, meantone does not split it at all, so we say it is ''monocot''. We also say it is ''haploid'', since it does not split~~ the octave ~~(2/1) at all~~.		Following intuitions from [[ploidacot]], one way to characterize temperaments is by how many parts they split important intervals into - for example, the perfect fifth and the octave.

	~~The elements~~ of a ~~wedgie each represent the number of parts into which a particular [[subgroup]] is split by the~~ temperament, or ~~the number~~ of ~~distinct sets~~ of ~~notes~~ within the temperament ~~linked~~ by ~~motions within~~ that subgroup. ~~They can be thought of as~~ a ~~generalization of the ploidacot information to all possible combinations of~~ [[~~prime harmonic~~\|~~primes~~]] ~~within~~ a temperament, and ~~format that information in a concise manner; as it turns out~~, this is enough to uniquely characterize the temperament.		For any n-prime subgroup of a rank-n temperament, there exist a finite number (1 or more) of copies of that subgroup within the temperament. Think of these as "universes" that are connected exclusively by intervals of that subgroup and may be travelled between by using intervals outside the subgroup. For example, a temperament that is [[Ploidacot/Diploid dicot\|diploid dicot]] - dividing the octave into two parts and also dividing the perfect fifth into two parts - will have a 2.3 wedgie entry of 4 (since there are four distinct copies of the 3-limit - the basic 3-limit, offset by a neutral third, offset by a semioctave, and offset by both). Each wedgie entry counts the number of copies of its corresponding subgroup. Because any temperament can be defined by splitting some interval and assigning the parts just interpretations, this is enough to uniquely characterize the temperament.

	For example, take the wedgie for meantone: {{multival\| 1 4 4 }}. Each entry corresponds to a pair of primes: 2.3, 2.5, and 3.5. The first entry of the wedgie is the ploidacot signatures multiplied together, which in this case is 1, ~~telling us that~~ the octave finds 2 at one step, and the fifth finds 3 at one generator minus one octave (which, since there are no even splits, still counts as 1).		=== Example 1: Meantone {{multival\| 1 4 4 }} ===
			For example, take the wedgie for meantone: {{multival\| 1 4 4 }}. Each entry corresponds to a pair of primes: 2.3, 2.5, and 3.5. The first entry of the wedgie is the ploidacot signatures multiplied together, which in this case is 1, because the octave finds 2 at one step, and the fifth finds 3 at one generator minus one octave (which, since there are no even splits, still counts as 1).
	For an intuition as to why multiplication is performed, consider [[hemipyth]] (or ''diploid dicot''), which divides both the octave and perfect fifth into two parts. In hemipyth, you can use [[radical interval\|radical]] steps to traverse between four different "universes" connected by 3-limit intervals: base Pythagorean, offset by a semioctave, offset by a neutral third, and offset by both. The wedgie entry essentially counts these universes.

	For 2.5, the procedure generalizes, with the entry, 4, being the number of steps 2 and 5 are divided into respectively multiplied together; that is, the number of parts into which the 2.5 subgroup is split. But since we already know 2 is divided into only one octave, this must mean 5 is split into four parts. In fact, 5 is found at four fifths up.		For 2.5, the procedure generalizes, with the entry, 4, being the number of steps 2 and 5 are divided into respectively multiplied together; that is, the number of parts into which the 2.5 subgroup is split. But since we already know 2 is divided into only one octave, this must mean 5 is split into four parts. In fact, 5 is found at four fifths up.
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	For the final entry, which is for the 3.5 subgroup, we have another 4. But this time, we are thinking tritave-equivalently now, so we will be reaching 5/3. This is the number of parts 3 and 5/3 are divided into respectively, multiplied together. 3 is reached by going up one 3/2 and one 2/1, but no splitting is happening, so the factor of 4 must come from 5/3, which is indeed reached by four 3/2's.		For the final entry, which is for the 3.5 subgroup, we have another 4. But this time, we are thinking tritave-equivalently now, so we will be reaching 5/3. This is the number of parts 3 and 5/3 are divided into respectively, multiplied together. 3 is reached by going up one 3/2 and one 2/1, but no splitting is happening, so the factor of 4 must come from 5/3, which is indeed reached by four 3/2's.

			=== Example 2: Father {{multival\| 1 -1 -4 }} ===
	For another example, take father, which has the wedgie {{multival\| 1 -1 -4 }}.		For another example, take father, which has the wedgie {{multival\| 1 -1 -4 }}.

	Here, we again have a 1 as our entry for 2.3, meaning that the temperament is haploid monocot, or in other words that 2/1 is unsplit and 3/2 is one generator.		Here, we again have a 1 as our entry for 2.3, meaning that the temperament is haploid monocot, or in other words that 2/1 is unsplit and 3/2 is one generator.

	Thus, going into our second entry, -1 for 2.5. Again, since we already know our temperament is haploid, the negative sign must come from the mapping for 5. But what could a negative sign possibly mean in a wedgie? Well, to reach the 5th harmonic (specifically, 5/4) in father, you go up a perfect ''fourth''. Since we are thinking octave-equivalently, this can be seen as going ''down'' a perfect fifth.		Thus, going into our second entry, -1 for 2.5. Again, since we already know our temperament is haploid, the negative sign must come from the mapping for 5. But what could a negative sign possibly mean in a wedgie? Well, to reach the 5th harmonic (specifically, 5/4) in father, you go up a perfect ''fourth''. Since we are thinking octave-equivalently, this can be seen as going ''down'' a perfect fifth. This means there are "-1 copies of 2.5" in father. This may seem weird, but {{multival\| 1 1 ...}} would imply that 5/4 is the same interval as 3/2, which is not what father does. Essentially, as part of the counting, two numbers are being multiplied together, and one of those numbers is negative when you go down instead of up to find the interval.

	Finally, for 3.5, we have the entry -4. Again, we are tritave-equivalent and 3/1 is simply found by an octave and a fifth, so we will be finding 5/3 by splitting it into four parts. 5/3 is equated to 16/9 in father, which is found by going up two octaves and down two fifths. This might seem like only a split into two, but keep in mind - we are in tritave-equivalent territory. Octaves are the tritave complement of fifths. So instead of going up two octaves, we can instead simply go down two more fifths to reach 5/3. And there we have it – 5/3 is split into four parts, which each contain a negative generator.		Finally, for 3.5, we have the entry -4. Again, we are tritave-equivalent and 3/1 is simply found by an octave and a fifth, so we will be finding 5/3 by splitting it into four parts. 5/3 is equated to 16/9 in father, which is found by going up two octaves and down two fifths. This might seem like only a split into two, but keep in mind - we are in tritave-equivalent territory. Octaves are the tritave complement of fifths. So instead of going up two octaves, we can instead simply go down two more fifths to reach 5/3. And there we have it – 5/3 is split into four parts, which each contain a negative generator.

			=== Example 3: Blackwood {{multival\| 0 5 8 }} ===
	For our final example, we will consider blackwood {{multival\| 0 5 8 }}.		For our final example, we will consider blackwood {{multival\| 0 5 8 }}.

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	And then the final entry, for 3.5, is 8. Again, 5 (i.e. 5/3) is found at one generator, but 3/1 is split into 8 parts by blackwood temperament. So, {{nowrap\| 1 × 8 {{=}} 8 }}.		And then the final entry, for 3.5, is 8. Again, 5 (i.e. 5/3) is found at one generator, but 3/1 is split into 8 parts by blackwood temperament. So, {{nowrap\| 1 × 8 {{=}} 8 }}.

	For wedgies of temperaments of larger prime subgroups, the number of entries is increased, so a rank-2 temperament of 7-limit JI would have 6 entries, for 2.3, 2.5, 2.7, 3.5, 3.7, and 5.7. Note that the new septimal entries are inserted between the entries for the 5-limit. If it helps, think of arranging all the entries in a grid, where rows represent the first prime, and columns represent the second, and reading them off one by one.		=== Generalizations ===
			For wedgies of temperaments of larger prime subgroups, the number of entries is increased, so a rank-2 temperament of 7-limit JI would have 6 entries, for 2.3, 2.5, 2.7, 3.5, 3.7, and 5.7. Note that the new septimal entries are inserted between the entries for the 5-limit (so that the 5-limit entries of a 7-limit wedgie are . If it helps, think of arranging all the entries in a grid, where rows represent the first prime, and columns represent the second, and reading them off one by one.

	For wedgies of higher-rank temperaments, the number of primes per entry is increased, so that for a rank-3 temperament of the 7-limit, all possible combinations of 3 primes (2.3.5, 2.3.7, 2.5.7, and 3.5.7) would be covered.		For wedgies of higher-rank temperaments, the number of primes per entry is increased, so that for a rank-3 temperament of the 7-limit, all possible combinations of 3 primes (2.3.5, 2.3.7, 2.5.7, and 3.5.7) would be covered.
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	== Wedgies and contorsion ==		== Wedgies and contorsion ==
	{{Todo\|inline=1\| expand }}		{{Todo\|inline=1\| expand }}

	== Conversion ==		== Conversion ==
	=== Mapping matrix to wedgie ===		=== Mapping matrix to wedgie ===

FloraC: /* Conversion */ comment

2025-08-30T13:37:33Z

Conversion: comment

← Older revision		Revision as of 13:37, 30 August 2025
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	# normalize for a positive first entry		# normalize for a positive first entry
			# unneeded if the mapping is in canonical form
	if wedgie[0] < 0:		if wedgie[0] < 0:
	wedgie *= -1		wedgie *= -1

FloraC: /* Derivation from mapping matrices */ expand. + quest for inverse operation

2025-08-30T13:12:09Z

Derivation from mapping matrices: expand. + quest for inverse operation

← Older revision		Revision as of 13:12, 30 August 2025
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	{{Todo\|inline=1\| expand }}		{{Todo\|inline=1\| expand }}

	== ~~Derivation from mapping matrices~~ ==		== Conversion ==
	{{~~Todo~~\|~~inline=1\| expand~~ }}		=== Mapping matrix to wedgie ===
			The wedgie may be found from the mapping by taking the {{w\|determinant}}s of the mapping's column slices that correspond to all the combinations of formal primes.

	Below is a ~~script that finds the wedgie from a mapping matrix in~~ [https://www.python.org/ Python], using [https://scipy.org/ Scipy].		Below is a [https://www.python.org/ Python] script that finds the wedgie from a mapping matrix, using [https://scipy.org/ Scipy].

	<syntaxhighlight lang="python">		<syntaxhighlight lang="python">
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	combinations = itertools.combinations (range (breeds.shape[1]), breeds.shape[0])		combinations = itertools.combinations (range (breeds.shape[1]), breeds.shape[0])
	wedgie = np.array ([linalg.det (breeds[:, entry]) for entry in combinations])		wedgie = np.array ([linalg.det (breeds[:, entry]) for entry in combinations])

			# normalize for a positive first entry
	if wedgie[0] < 0:		if wedgie[0] < 0:
	wedgie *= -1		wedgie *= -1
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	return wedgie		return wedgie
	</syntaxhighlight>		</syntaxhighlight>

			=== Wedgie to mapping matrix ===
			{{Todo\|inline=1\| expand }}

	== Derivation from edo joins ==		== Derivation from edo joins ==

FloraC: /* Derivation from mapping matrices */ + Python script

2025-08-21T11:41:54Z

Derivation from mapping matrices: + Python script

@@ Line 70: / Line 70: @@
 == Derivation from mapping matrices ==
 {{Todo|inline=1| expand }}
 == Derivation from edo joins ==

VectorGraphics at 07:49, 21 August 2025

2025-08-21T07:49:08Z

← Older revision		Revision as of 07:49, 21 August 2025
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	== Derivation from edo joins ==		== Derivation from edo joins ==
	{{See also\|Wedge product }}		{{See also\|Wedge product }}
			[[File:Wedgediagram .png\|thumb\|504x504px\|Wedge product diagram]]
	Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging {{val\| 5 8 12 }} and {{val\| 7 11 16 }} (the patent vals for 5edo and 7edo) yields {{multival\| (5×11 - 8×7) (5×16 - 12×7) (8×16 - 12×11) }}, which simplifies to {{multival\| (55 - 56) (80 - 84) (128 - 132) }} and thus to {{multival\| -1 -4 -4 }}. Note that we generally assume the first entry of the wedgie should be positive, for which we flip all the signs of it to obtain {{multival\| 1 4 4 }}, which is the wedgie for 5 & 7, a.k.a. meantone.		Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging {{val\| 5 8 12 }} and {{val\| 7 11 16 }} (the patent vals for 5edo and 7edo) yields {{multival\| (5×11 - 8×7) (5×16 - 12×7) (8×16 - 12×11) }}, which simplifies to {{multival\| (55 - 56) (80 - 84) (128 - 132) }} and thus to {{multival\| -1 -4 -4 }}. Note that we generally assume the first entry of the wedgie should be positive, for which we flip all the signs of it to obtain {{multival\| 1 4 4 }}, which is the wedgie for 5 & 7, a.k.a. meantone.

VectorGraphics at 07:22, 21 August 2025

2025-08-21T07:22:09Z

← Older revision		Revision as of 07:22, 21 August 2025
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	== Derivation from edo joins ==		== Derivation from edo joins ==
	{{See also\| ~~Linear algebra formalism #~~Wedge product }}		{{See also\|Wedge product }}

	Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging {{val\| 5 8 12 }} and {{val\| 7 11 16 }} (the patent vals for 5edo and 7edo) yields {{multival\| (5×11 - 8×7) (5×16 - 12×7) (8×16 - 12×11) }}, which simplifies to {{multival\| (55 - 56) (80 - 84) (128 - 132) }} and thus to {{multival\| -1 -4 -4 }}. Note that we generally assume the first entry of the wedgie should be positive, for which we flip all the signs of it to obtain {{multival\| 1 4 4 }}, which is the wedgie for 5 & 7, a.k.a. meantone.		Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging {{val\| 5 8 12 }} and {{val\| 7 11 16 }} (the patent vals for 5edo and 7edo) yields {{multival\| (5×11 - 8×7) (5×16 - 12×7) (8×16 - 12×11) }}, which simplifies to {{multival\| (55 - 56) (80 - 84) (128 - 132) }} and thus to {{multival\| -1 -4 -4 }}. Note that we generally assume the first entry of the wedgie should be positive, for which we flip all the signs of it to obtain {{multival\| 1 4 4 }}, which is the wedgie for 5 & 7, a.k.a. meantone.

FloraC: Note the first entry should be positive

2025-06-29T10:46:23Z

Note the first entry should be positive

← Older revision		Revision as of 10:46, 29 June 2025
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	{{See also\| Linear algebra formalism #Wedge product }}		{{See also\| Linear algebra formalism #Wedge product }}

	Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging {{val\| 5 8 12 }} and {{val\| 7 11 16 }} (the patent vals for 5edo and 7edo) yields {{multival\| (5×11 - 8×7) (5×16 - 12×7) (8×16 - 12×11) }}, which simplifies to {{multival\| (55 - 56) (80 - 84) (128 - 132) }} and thus to {{multival\| -1 -4 -4 }}~~, which is the wedgie for 5 & 7, a.k.a. meantone~~. Note that ~~if you take a~~ wedgie ~~and~~ flip all the signs, it ~~still results in the same wedgie, so~~ {{multival\| -1 -4 -4 }} is the ~~same as {{multival\| 1 4 4 }} from before~~.		Two [[vals]] can be combined into a wedgie representing the rank-2 temperament they both support using the wedge product. For example, wedging {{val\| 5 8 12 }} and {{val\| 7 11 16 }} (the patent vals for 5edo and 7edo) yields {{multival\| (5×11 - 8×7) (5×16 - 12×7) (8×16 - 12×11) }}, which simplifies to {{multival\| (55 - 56) (80 - 84) (128 - 132) }} and thus to {{multival\| -1 -4 -4 }}. Note that we generally assume the first entry of the wedgie should be positive, for which we flip all the signs of it to obtain {{multival\| 1 4 4 }}, which is the wedgie for 5 & 7, a.k.a. meantone.

	More than two vals can be combined into a higher-rank wedgie by an analogous method.		More than two vals can be combined into a higher-rank wedgie by an analogous method.

FloraC: If something "takes some form", it should cover all the possible cases

2025-06-29T10:43:00Z

If something "takes some form", it should cover all the possible cases

← Older revision		Revision as of 10:43, 29 June 2025
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	{{Beginner\|Plücker coordinates}}		{{Beginner\|Plücker coordinates}}

	A '''wedgie''' is an object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] vs. [[generator]] or of [[equave]], which can therefore provide some illuminating information beyond the [[mapping\|mapping matrix]], which specifies a particular equave. A wedgie takes the form ~~{{multival\| x y z }}~~, with ~~a number of~~ entries ~~(not necessarily 3; in fact that is the simplest possible case)~~ listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, ~~etc.~~). Wedgies can be thought of as a generalization of vals, called ''multivals'', so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.		A '''wedgie''' is an object that uniquely characterizes a [[regular temperament]] regardless of choice of [[period]] vs. [[generator]] or of [[equave]], which can therefore provide some illuminating information beyond the [[mapping\|mapping matrix]], which specifies a particular equave. A wedgie takes the form ⟨⟨…⟨ ''w''<sub>1</sub> ''w''<sub>2</sub> … ''w''<sub>''n''</sub> ]]…], with ''n'' entries listed in between multiple [[val]] brackets (double brackets for rank-2, triple brackets for rank-3, and so on). Wedgies can be thought of as a generalization of vals, called ''multivals'', so that a val is a wedgie for a rank-1 temperament. Each element conveys information about the structure of a set of primes in the temperament, containing a number of primes equivalent to the temperament's rank.

	== How to read a wedgie ==		== How to read a wedgie ==